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December, 1979 An Almost Sure Invariance Principle for the Partial Sums of Infima of Independent Random Variables
H. Hebda-Grabowska, D. Szynal
Ann. Probab. 7(6): 1036-1045 (December, 1979). DOI: 10.1214/aop/1176994896

Abstract

Let $\{X_n, n \geqslant 1\}$ be a sequence of independent random variables uniformly distributed on the unit interval. Put $X^\ast_n = \inf(X_1, X_2,\cdots, X_n)$ and $S_n = X^\ast_1 + X^\ast_2 + \cdots + X^\ast_n, n \geqslant 2, S_1 = 0$. The aim of this note is to give an almost sure invariance principle for $S_n$. Next we extend the given results to the case when $X_n, n \geqslant 1$, are not uniformly distributed but bounded, and moreover, to sums $\hat{S}_n = X^{(m)}_m + X^{(m)}_{m+1} +\cdots + X^{(m)}_n$, where $X^{(m)}_j$ is the $m$th order statistic of $(X_1, X_2,\cdots, X_j)$.

Citation

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H. Hebda-Grabowska. D. Szynal. "An Almost Sure Invariance Principle for the Partial Sums of Infima of Independent Random Variables." Ann. Probab. 7 (6) 1036 - 1045, December, 1979. https://doi.org/10.1214/aop/1176994896

Information

Published: December, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0423.60031
MathSciNet: MR548897
Digital Object Identifier: 10.1214/aop/1176994896

Subjects:
Primary: 60B10
Secondary: 60F15

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 6 • December, 1979
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